curl: Vector Field Flow Analysis

Advanced Flow Pattern Generation Through Differential Calculus

The curl function computes the rotational characteristics of vector fields generated from noise functions. This mathematical operator calculates the circulation density at each point in a field, creating natural swirling and turbulent patterns essential for fluid dynamics simulation.

Mathematical Foundation

For a 2D vector field $\vec{F}(x,y) = (P(x,y), Q(x,y))$, the curl operation is defined as:

$$\text{curl } \vec{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$$

For 3D vector fields $\vec{F}(x,y,z) = (P, Q, R)$, the curl becomes:

$$\nabla \times \vec{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)$$

The implementation uses finite differences with step size $\epsilon = 0.1$ to approximate these partial derivatives numerically.

Function Variants

Function	Input Type	Output Type	Purpose
curlVec2	vec2	vec2	2D flow field analysis
curlVec3	vec3	vec3	3D turbulence patterns
curlVec4	vec4	vec3	Time-varying flow evolution

Implementation

Live Editor

const fragment = () => {
      const p = uv.mul(4)
      const flow = curlVec2(p.add(iTime.mul(0.1)))
      const angle = flow.y.atan2(flow.x).add(float(3.14159)).div(float(3.14159).mul(2))
      return vec4(flow, angle, 1)
}

Live Editor

const fragment = () => {
      const p = vec3(uv.mul(3), iTime.mul(0.2))
      const vorticity = curlVec3(p)
      const magnitude = vorticity.length()
      const color = vorticity.abs().normalize().mul(magnitude).pow(vec3(0.7))
      return vec4(color, 1)
}